2014
DOI: 10.1093/imrn/rnu033
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Extensions of Buchi's Higher Powers Problem to Positive Characteristic

Abstract: Büchi's n-th power problem on Q asks whether there exist an integer M such that the only monic polynomials F ∈ Q[X] of degree n satisfying that F (1),. .. , F (M) are n-th power rational numbers, are precisely of the form F (X) = (X +c) n for some c ∈ Q. In this paper, we study analogues of this problem for algebraic function fields of positive characteristic. We formulate and prove an analogue (indeed, such a formulation for n > 2 was missing in the literature due to some unexpected phenomena), which we use t… Show more

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Cited by 5 publications
(5 citation statements)
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“…Analogues of Question 1.1 have been considered for most classical rings (but in the case of number fields, under some well-known conjectures, like Bombieri-Lang for surfaces, or some version of ABC). Relevant results in positive characteristic can be found in [Pa11] (the analogue of Büchi's problem for any power over fields with a prime number of elements), and in [PaW15] (over rings of functions), who generalize previous results in [PhV06,PhV10,ShV10,AW11,AHW13]. For a general survey on Büchi's problem and its extensions to other structures and higher powers, see [PaPhV10].…”
Section: Introductionmentioning
confidence: 64%
“…Analogues of Question 1.1 have been considered for most classical rings (but in the case of number fields, under some well-known conjectures, like Bombieri-Lang for surfaces, or some version of ABC). Relevant results in positive characteristic can be found in [Pa11] (the analogue of Büchi's problem for any power over fields with a prime number of elements), and in [PaW15] (over rings of functions), who generalize previous results in [PhV06,PhV10,ShV10,AW11,AHW13]. For a general survey on Büchi's problem and its extensions to other structures and higher powers, see [PaPhV10].…”
Section: Introductionmentioning
confidence: 64%
“…Let us recall the following theorem from [17] (see also [16], [20] and [13]). It gives a solution to Büchi's n squares problem for polynomial rings in positive characteristic.…”
Section: Büchi's Problem and The P R -Th Power Relationmentioning
confidence: 99%
“…Our proof of Theorem 2 contains two major ingredients, both rely on the special features of function fields of characteristic zero. One of the ingredients is the result (restated as Theorem 23 in Section 4) by Pasten and Wang [12] motivated by Büchi's d-th power problem, which has a similar flavor as Pisot's d-th root conjecture but arising from different purposes. While working on an undecidability problem related to Hilbert's tenth problem in the 1970s, Büchi formulated a related arithmetic problem, which can be stated in more generality as follows: Let k be a number field.…”
Section: Introductionmentioning
confidence: 99%
“…In particular, the analog over function fields of characteristic zero was solved completely, even with an explicit bound on M ; see [2] and [11]. We refer to [12] for a survey of relevant works. The other ingredient, which is also developed in this paper, is the function-field analog of the recent work of Levin [7] for number fields and Levin-Wang [8] for meromorphic functions on GCD estimates of two multivariable polynomials over function fields evaluated at arguments which are S-units.…”
Section: Introductionmentioning
confidence: 99%
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